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Theorem isnzr 17353
Description: Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Hypotheses
Ref Expression
isnzr.o  |-  .1.  =  ( 1r `  R )
isnzr.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
isnzr  |-  ( R  e. NzRing 
<->  ( R  e.  Ring  /\  .1.  =/=  .0.  )
)

Proof of Theorem isnzr
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 fveq2 5703 . . . 4  |-  ( r  =  R  ->  ( 1r `  r )  =  ( 1r `  R
) )
2 isnzr.o . . . 4  |-  .1.  =  ( 1r `  R )
31, 2syl6eqr 2493 . . 3  |-  ( r  =  R  ->  ( 1r `  r )  =  .1.  )
4 fveq2 5703 . . . 4  |-  ( r  =  R  ->  ( 0g `  r )  =  ( 0g `  R
) )
5 isnzr.z . . . 4  |-  .0.  =  ( 0g `  R )
64, 5syl6eqr 2493 . . 3  |-  ( r  =  R  ->  ( 0g `  r )  =  .0.  )
73, 6neeq12d 2635 . 2  |-  ( r  =  R  ->  (
( 1r `  r
)  =/=  ( 0g
`  r )  <->  .1.  =/=  .0.  ) )
8 df-nzr 17352 . 2  |- NzRing  =  {
r  e.  Ring  |  ( 1r `  r )  =/=  ( 0g `  r ) }
97, 8elrab2 3131 1  |-  ( R  e. NzRing 
<->  ( R  e.  Ring  /\  .1.  =/=  .0.  )
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2618   ` cfv 5430   0gc0g 14390   1rcur 16615   Ringcrg 16657  NzRingcnzr 17351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-rex 2733  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-iota 5393  df-fv 5438  df-nzr 17352
This theorem is referenced by:  nzrnz  17354  nzrrng  17355  drngnzr  17356  isnzr2  17357  rngelnzr  17359  subrgnzr  17361  chrnzr  17973  nrginvrcn  20284  ply1nzb  21606  zrhnm  26410  isdomn3  29584  isnzr2hash  30786
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